A Gleason Solution Model for Row Contractions

Abstract

In the de Branges–Rovnyak functional model for contractions on Hilbert space, any completely non-coisometric (CNC) contraction is represented as the adjoint of the restriction of the backward shift to a de Branges– Rovnyak space, \(\mathcal{H}(b)\), associated to a contractive analytic operator-valued function, b, on the open unit disk.

We extend this model to a large class of CNC contractions of several copies of a Hilbert space into itself (including all CNC row contractions with commuting component operators). Namely, we completely characterize the set of all CNC row contractions, T , which are unitarily equivalent to an extremal Gleason solution for a de Branges–Rovnyak space, \(\mathcal{H}(b_{T})\), contractively contained in a vector-valued Drury–Arveson space of analytic functions on the open unit ball in several complex dimensions. Here, a Gleason solution is the appropriate several-variable analogue of the adjoint of the restricted backward shift and the characteristic function, bT, belongs to the several-variable Schur class of contractive multipliers between vector-valued Drury–Arveson spaces. The characteristic function, bT, is a unitary invariant, and we further characterize a natural sub-class of CNC row contractions for which it is a complete unitary invariant.